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EDIT: A simple proof of Lemma 6. The group $N(K^s)$ acts on $\X_*(T)$, $\X_*(T^\ssc)$, and $\pi_1(G,T)$ via the Weyl group $W=W(G,T)=N(K^s)/T(K^s)$.

Let $\X^*(T)$ denote the character group of $T$ (over $K^s$). Let $R=R(G,T)\subset \X^*(T)$ and $R^\vee=R^\vee(G,T)\subset X_*(T)$ denote the corresponding root and coroot systems. Then $W$ is generated by the reflections $s_\alpha$ for $\alpha\in R$. It suffices to show that each $s_\alpha$ acts trivially on $\pi_1(G,T)$.

Let $\alpha^\vee\in R^\vee$ denote the coroot corresponding to the root $\alpha$. Then the reflection $s_\alpha$ acts on $\X_*(T)$ by $$ s_\alpha(u)= u-\langle\alpha,u\rangle \alpha^\vee $$ for $u\in \X_*(T)$. See Section 1.1 in Springer, Reductive groups, Proc. Sympos. Pure Math., XXXIII, American Mathematical Society, Providence, RI, 1979, Part 1, pp. 3–27. Since $\alpha^\vee\in R^\vee\subset \rho_*\X_*(T^\ssc)$, we see that $s_\alpha$ indeed acts trivially on $\pi_1(G,T)=\X_*(T)/\X_*\rho_*(T^\ssc)$. This completes the proof of Lemma 6.

Let $N_H$ denote the normalizer of $T_H$ in $H$. We consider the Weyl group $W=N_H/T_H\cong N/T$, which naturally and compatibly acts on $\pi_1(H,T_H)$ and on $\pi_1(G,T)$ Since the torus $C_H$ is central in $H$, we see that $N_H$ acts trivially on $C_H$ and on $ \X_*(C_H)\otimes \Q$. Therefore, it acts trivially on $\X_*(H^\tor)\otimes \Q$ and on $ \X_*(H^\tor)$. Thus $W$ acts trivially on $\pi_1(H,T_H)=\X_*(H^\tor)$, and we see from \eqref{e:4} that $W$ acts trivially on $\pi_1(G,T)$. This completes the proofs of Lemma 6, Lemma 3, Corollary 4, and Proposition 1.

Let $N_H$ denote the normalizer of $T_H$ in $H$. We consider the Weyl group $W=N_H/T_H\cong N/T$, which naturally and compatibly acts on $\pi_1(H,T_H)$ and on $\pi_1(G,T)$. Since the torus $C_H$ is central in $H$, we see that $N_H$ acts trivially on $C_H$ and on $ \X_*(C_H)\otimes \Q$. Therefore, it acts trivially on $\X_*(H^\tor)\otimes \Q$ and on $ \X_*(H^\tor)$. Thus $W$ acts trivially on $\pi_1(H,T_H)=\X_*(H^\tor)$, and we see from \eqref{e:4} that $W$ acts trivially on $\pi_1(G,T)$. This completes the proofs of Lemma 6, Lemma 3, Corollary 4, and Proposition 1.

Notation. We write $G^\sss=[G,G]$ for the derived group of $G$ (it is semisimple), and $G^\tor=G/G^\sss$ (it is a $K$-torus). We denote by $G^\ssc$ the universal cover of $G^\sss$ (it is simply connected) and consider the composite homomorphism $$ \rho\colon G^\ssc\twoheadrightarrow G^\sss\hookrightarrow G.$$

Proposition 1. For any two maximal tori $T_1,T_2\subseteq G$, there is a canonical isomorphism $$\varphi_{12}\colon \pi_1(G_1,T_1)\overset\sim\longrightarrow \pi_1(G_2,T_2).$$ Moreover, for any third maximal torus $T_3\subseteq G$, we have $$\varphi_{13}=\varphi_{23}\circ\varphi_{12}$$ with the obvious notations.

Notation. We write $G^\sss=[G,G]$ for the derived group of $G$ (it is semisimple), and $G^\tor=G/G^\sss$ (it is a $K$-torus). We denote by $G^\ssc$ the universal cover of $G^\sss$ (it is simply connected) and consider the composite homomorphism $$ \rho\colon\, G^\ssc\twoheadrightarrow G^\sss\hookrightarrow G.$$

Proposition 1. For any two maximal tori $T_1,T_2\subseteq G$, there is a canonical isomorphism of $\Gamma$-modules $$\varphi_{12}\colon\, \pi_1(G_1,T_1)\overset\sim\longrightarrow \pi_1(G_2,T_2).$$ Moreover, for any third maximal torus $T_3\subseteq G$, we have $$\varphi_{13}=\varphi_{23}\circ\varphi_{12}$$ with the obvious notations.